Is that series $\sum_{k=1}^{\infty}\frac{z^k}{ak + 1}$ related to a common function?

Question

I have the series

$$\sum_{k=1}^{\infty}\frac{z^k}{ak + 1}$$

and am wondering whether it relates to any function that any of you knows?

The most similar that I know is the series representation of a polylogarithm of order $s$ is given by

$$\text{Li}_s(z) = \sum_{k=1}^{\infty}\frac{z^k}{k^s}$$

with $s=1$ but I cannot find anything closer.

Answer 1

The series (starting at $k=0$) is $a^{-1} \Phi(z, 1, 1/a)$ where $\Phi$ is the Lerch Phi function.

Answer 2

If a is an integer, this will be a multisection of $\ln(x)$.